What is a Hypergraph? What is Hypergraph Partitioning?

Hypergraphs are a generalization of graphs, where each (hyper)edge (also called net) can
connect more than two vertices. The k-way hypergraph partitioning problem is the generalization of the well-known graph partitioning problem: partition the vertex set into k disjoint
blocks of bounded size (at most 1 + ε times the average block size), while minimizing an
objective function defined on the nets.

The two most prominent objective functions are the cut-net and the connectivity (or λ − 1)
metrics. Cut-net is a straightforward generalization of the edge-cut objective in graph partitioning
(i.e., minimizing the sum of the weights of those nets that connect more than one block). The
connectivity metric additionally takes into account the actual number λ of blocks connected by a
net. By summing the (λ − 1)-values of all nets, one accurately models the total communication
volume of parallel sparse matrix-vector multiplication and once more gets a metric that reverts
to edge-cut for plain graphs.

What is KaHyPar?

KaHyPar is a multilevel hypergraph partitioning framework for optimizing the cut- and the
(λ − 1)-metric. It supports both recursive bisection and direct k-way partitioning.
As a multilevel algorithm, it consist of three phases: In the coarsening phase, the
hypergraph is coarsened to obtain a hierarchy of smaller hypergraphs. After applying an
initial partitioning algorithm to the smallest hypergraph in the second phase, coarsening is
undone and, at each level, a local search method is used to improve the partition induced by
the coarser level. KaHyPar instantiates the multilevel approach in its most extreme version,
removing only a single vertex in every level of the hierarchy.
By using this very fine grained n-level approach combined with strong local search heuristics,
it computes solutions of very high quality.
Its algorithms and detailed experimental results are presented in several research publications.

Additional Features

Hypergraph Partitioning with Variable Block Weights:

KaHyPar has support for variable block weights. If command line option --use-individual-part-weights=true is used, the partitioner tries to partition the hypergraph such that each block Vx has a weight of at most Bx, where Bx can be specified for each block individually using the command line parameter --part-weights= B1 B2 B3 ... Bk-1. Since the framework does not yet support perfectly balanced partitioning, upper bounds need to be slightly larger than the total weight of all vertices of the hypergraph. Note that this feature is still experimental.

Hypergraph Partitioning with Fixed Vertices:

Hypergraph partitioning with fixed vertices is a variation of standard hypergraph partitioning. In this problem, there is an additional constraint on the block assignment of some vertices, i.e., some vertices are preassigned to specific blocks prior to partitioning with the condition that, after partitioning the remaining “free” vertices, the fixed vertices are still in the block that they were assigned to. The command line parameter --fixed / -f can be used to specify a fix file in hMetis fix file format. For a hypergraph with V vertices, the fix file consists of

V

lines - one for each vertex. The ith line either contains -1 to indicate that the vertex is free to move or <part id> to indicate that this vertex should be preassigned to block <part id>. Note that part ids start from 0.

KaHyPar currently supports three different contraction policies for partitioning with fixed vertices:

free_vertex_only allows all contractions in which the contraction partner is a free vertex, i.e., it allows contractions of vertex pairs where either both vertices are free, or one vertex is fixed and the other vertex is free.

fixed_vertex_allowed additionaly allows contractions of two fixed vertices provided that both are preassigned to the same block. Based on preliminary experiments, this is currently the default policy.

equivalent_vertices only allows contractions of vertex pairs that consist of either two free vertices or two fixed vertices preassigned to the same block.

Evolutionary Framework (KaHyPar-E):

KaHyPar-E enhances KaHyPar with an evolutionary framework as described in our GECCO’18 publication. Given a fairly large amount of running time, this memetic multilevel algorithm performs better than repeated executions of KaHyPar-MF/-CA, hMetis, and PaToH. The configuration /config/km1_direct_kway_gecco18.ini uses KaHyPar-CA to exploit the local solution space and was used in the GECCO’18 experiments. The command line parameter --time-limit=xxx can be used to set the maximum running time (in seconds). Parameter --partition-evolutionary=true enables evolutionary partitioning.

Experimental Results

We use the performance plots introduced in ALENEX’16 to compare KaHyPar to other partitioning algorithms in terms of solution quality:
For each algorithm, these plots relate the smallest minimum cut of all algorithms to the
corresponding cut produced by the algorithm on a per-instance basis. For each algorithm,
these ratios are sorted in increasing order. The plots use a cube root scale for both axes
to reduce right skewness and show 1 − (best/algorithm) on the y-axis to highlight the
instances were each partitioner performs badly. A point close to one indicates that the
partition produced by the corresponding algorithm was considerably worse than the partition
produced by the best algorithm. A value of zero therefore indicates that the corresponding
algorithm produced the best solution. Points above one correspond to infeasible solutions
that violated the balance constraint. Thus an algorithm is considered to outperform another
algorithm if its corresponding ratio values are below those of the other algorithm.

Performance plots and detailed per-instance results can be found on
the website accompanying each publication.

Testing and Profiling

Tests are automatically executed while project is built. Additionally a test target is provided.
End-to-end integration tests can be started with: make integration_tests. Profiling can be enabled via cmake flag: -DENABLE_PROFILE=ON.

Running KaHyPar

The binary is located at: build/kahypar/application/.

KaHyPar has several configuration parameters. For a list of all possible parameters please run: ./KaHyPar --help.
We use the hMetis format for the input hypergraph file as well as the partition output file.

Note that the configuration km1_direct_kway_gecco18.ini is based on KaHyPar-CA. However, KaHyPar-E also works with flow-based local improvements if the configration is adjusted according to the refinement parameters used in km1_direct_kway_sea18.ini.

Bug Reports

Licensing

KaHyPar is free software provided under the GNU General Public License (GPLv3).
For more information see the COPYING file.
We distribute this framework freely to foster the use and development of hypergraph partitioning tools.
If you use KaHyPar in an academic setting please cite the appropriate paper. If you are interested in a commercial license, please contact me.

KaHyPar-MF integrates implementations of the BK and incremental breadth first search (IBFS) maximum flow algorithm into the framework (see /external_tools/maximum_flow/). The BK algorithm has been described in